Frege’s Recipe∗1

2

This paper has three aims: first, we present a formal recipe that Frege3

followed in his magnum opus Grundgesetze der Arithmetik1 when formulating4

his definitions. This generalized recipe, as we will call it, is not explicitly5

mentioned as such by Frege, but we will offer strong reasons to believe that6

Frege applied the recipe in developing the formal material of Grundgesetze.7

Second, we will show that a version of Basic Law V plays a fundamental role8

in the generalized recipe. We will explicate exactly what this role is and how it9

differs from the role played by extensions in Die Grundlagen der Arithmetik.210

Third, and finally, we will demonstrate that this hitherto neglected yet11

foundational aspect of Frege’s use of Basic Law V helps to resolve a number12

of important interpretative challenges in recent Frege scholarship, while also13

shedding light on some important differences between Frege’s logicism and14

recent neo-logicist approaches to the foundations of mathematics.15

The structure of our paper is as follows: In the first section, we will16

outline Frege’s semi-formal definition of cardinal numbers given in Grundlagen17

and present what we call the simple recipe. In the second section, we will18

outline two distinct ways to unpack the simple recipe formally, followed19

by a discussion of its philosophical and technical shortcomings. This leads20

naturally to the topic of the third section—the problem of the singleton—a21

problem that Frege was aware of and which, we believe, significantly shaped22

his views on definitions between Grundlagen and Grundgesetze. These23

observations motivate the introduction of the generalized recipe. In the24

∗We would like to thank the audience of an Early Analytic Group meeting at theUniversity of Stirling and the audience of the MidWestPhilMathWorkshop 13 for valuablefeedback. Furthermore, we would like to thank two anonymous referees for their extensiveand challenging comments which led to numerous improvements. This work was supportedby a grant awarded to Philip Ebert from the UK Arts and Humanities Research Council:AH/J00233X/1.

1Published in two volumes: Gottlob Frege, Grundgesetze der Arithmetik. Begriffss-chriftlich abgeleitet. vol. I. (Jena: Verlag H. Pohle, 1893) and Gottlob Frege, Grundgesetzeder Arithmetik. Begriffsschriftlich abgeleitet. vol. II. (Jena: Verlag H. Pohle, 1903), hence-forth: Grundgesetze. We follow the English translation by Philip A. Ebert and MarcusRossberg, trans., Gottlob Frege: Basic Laws of Arithmetic (Oxford: Oxford UniversityPress, 2013).

2Gottlob Frege, Die Grundlagen der Arithmetik. Eine logisch mathematische Un-tersuchung uber den Begriff der Zahl (Breslau: Wilhelm Koebner, 1884), henceforth:Grundlagen. English translation of the quoted passages were provided by the authors.

1

fourth section, we will explain this generalized modification of the simple25

recipe and demonstrate how it is applied in arriving at the majority of the26

definitions given in Grundgesetze. In the fifth section, we will argue that27

the generalized recipe has important philosophical consequences for Frege28

scholarship: we will sketch the beginnings of a new interpretation of the role29

and importance of Basic Law V and of Hume’s Principle in Frege’s mature30

(Grundgesetze-era) philosophy of mathematics. We close by noting some31

differences between Frege’s project and the methodology of contemporary32

neo-logicism.33

34

i. identifying abstracts in grundlagen35

As is well known, in Grundlagen Frege rejected Hume’s Principle:336

HP : (∀X)(∀Y )[”(X) = ”(Y )↔ X ≈ Y ]

as a definition of the concept cardinal number.4 Hume’s Principle states37

that the number of F ’s is identical to the number of G’s if and only if the38

F ’s and the G’s are in one-to-one correspondence. Frege was very likely39

aware of the fact that Hume’s Principle on its own (plus straightforward40

definitions of arithmetical concepts such as successor, addition, and41

multiplication) entails what we now call the second-order Dedekind-Peano42

axioms for arithmetic—a result that is known as Frege’s Theorem.5 The43

3Frege never used the expression “Hume’s Principle”. The use of this label is, however,entrenched amongst Frege scholars and so we will refer to this principle throughout using“Hume’s Principle”, even in a context when we discuss Frege’s views about it.

4“”” is the cardinal-number operator, and “X ≈ Y ” abbreviates the second-orderformula stating that there is a one-to-one onto function from X to Y. We shall partlytranslate Frege’s Grundgesetze formulations into modern terminology—with appropriatecomments regarding any theoretical mutilations that might result—but we will retainhis original notation in quotations. Hence, we will use modern ‘Australian’ A’s (∀) and‘backwards’ E’s (∃) for the quantifiers, and modern linear notation (→) for the materialconditional. The reader should be aware that identity (=) and equivalence (↔) are,within Frege’s Grundgesetze formalism, equivalent when the arguments are sentences (thatis, names of truth-values), and we will use whichever is more illuminating in our ownformulations below.

5The label “Frege’s Theorem” dates back to George Boolos “The Standard of Equalityof Numbers,” in George Boolos, ed. Meaning and Method: Essays in Honor of HilaryPutnam (Cambridge: Cambridge University Press, 1990), pp. 261-278. See also RichardG. Heck, Jr. Frege’s Theorem, (Oxford: Oxford University Press, 2011), p. 3ff on thehistorical context surrounding Frege’s Theorem. Dummett suggests that Frege was awarethat Peano Arithmetic could be derived solely from Hume’s Principle at the time of writingGrundlagen, see Michael Dummett Frege: Philosophy of Mathematics (Cambridge: HarvardUniversity Press, 1991), p. 123. However, as shown by Boolos and Heck, Frege’s sketchof this result—in particular, the proof sketch of the successor axiom—in Grundlagen is

2

reason for his rejection of Hume’s Principle as a proper definition is known44

as the Caesar Problem:45

...we can never—to take a crude example—decide by means of46

our definitions whether a concept has the number Julius Caesar47

belonging to it, whether this famous conqueror of Gaul is a48

number or not.649

The worry, in short, is that an adequate definition of the concept cardinal50

number should settle all identities involving numerical terms, including51

those where the identity symbol ‘=’ is flanked by a Fregean numeral (such52

as ‘”(F )’) on one side and a non-numerical term (such as ‘Julius Caesar’)53

on the other. Hume’s Principle does not settle such identities and thus it is54

inadequate as a definition of the concept cardinal number.55

Frege’s proposed solution to the Caesar Problem is simple to state: in56

order to distinguish numbers from more “pedestrian” objects, such as the57

conqueror of Gaul, Frege proposes that we identify cardinal numbers with58

certain extensions by means of an explicit definition. In Grundlagen, §68,59

immediately after a discussion of the Caesar Problem, Frege offers the60

following definition:61

Accordingly, I define:62

the cardinal number which belongs to the concept F is the ex-63

tension of the concept “equinumerous to the concept F”.764

Thus, cardinal numbers are a particular kind of extension. It is clear from65

his discussion in Grundlagen, however, that it is not just cardinal numbers66

but many (if not all) other mathematical objects that are to be identified67

with appropriate extensions. In the same section, he writes:68

the direction of line a is the extension of the concept “parallel to69

line a”70

the shape of triangle t is the extension of the concept “similar to71

the triangle t”.872

The wide-ranging nature of these examples strongly suggests that Frege73

regarded this approach not merely as a technical fix to resolve particular74

cases involving Caesar-type examples, but rather as a codification of a75

basic insight into the nature of mathematical objects and mathematical76

concepts. Frege’s identification of mathematical objects with the extension77

incorrect, see George Boolos and Richard G. Heck, Jr. “Die Grundlagen der Arithmetik§§82–83,” in Matthias Schirn, ed., Philosophy of Mathematics Today (Oxford: OxfordUniversity Press, 1998), pp. 407-428, see also Heck, Frege’s Theorem, pp.69-89.

6Frege, Grundlagen der Arithmetik, p. 68.7Frege, Grundlagen der Arithmetik, pp. 79-80.8Frege, Grundlagen der Arithmetik, p. 79.

3

of corresponding equivalence classes amounts to a definitional method which78

seems generally applicable to all mathematical objects and concepts, including79

shapes, directions, and cardinal numbers.9 Hence, from this perspective80

there is nothing special about cardinal numbers—they are just a particularly81

salient example of the definitional methodology applied in Grundlagen.1082

Reflecting on Frege’s methodology in Grundlagen, we obtain the following83

recipe for identifying the mathematical objects falling under some mathe-84

matical concept C (such as direction or shape), which we shall call the85

simple abstracta-as-extension recipe, or simply, the simple recipe:1186

Step 1: Identify the underlying concept ΦC such that C’s are C’s87

of ΦC ’s.88

That is, if C is the concept direction, then ΦC is the concept line, and if89

C is the concept shape, then ΦC is the concept triangle.90

Step 2: Formulate the identity conditions for C’s in terms of some91

appropriate equivalence relation ΨC on the underlying domain92

of ΦC ’s.93

That is, identify a formula of the form:94

∀φ1, φ2 ∈ ΦC , the C of φ1 = the C of φ2 ↔ ΨC(φ1, φ2)

where ΨC provides the identity conditions for C’s. Thus, if C is the concept95

direction, then ΨC is the relation parallelism, and if C is the concept96

shape, then ΨC is the relation similarity.97

Step 3: Identify the C’s with the equivalence classes of relevant98

ΦC ’s (modulo the equivalence relation ΨC).99

9There is, of course, Grundlagen §107, where Frege suggests that he attaches noparticular importance to his use of the term “extensions of concepts”. By the time ofGrundgesetze, however, Frege attaches a great deal of importance to extensions of concepts,or more generally, value-ranges of functions. This reflects a deep change in Frege’s viewsbetween the time of writing Grundlagen and Grundgesetze, one intimately connected tohis abandoning the simple recipe in favor of the generalized recipe. More on this below.

10There is, of course, something special about cardinal numbers when compared toshapes and directions: cardinal numbers are defined as extensions of second-level conceptsthat hold of concepts (or, alternatively, of first-level concepts that hold of extensions ofconcepts). Thus, cardinal numbers, unlike shapes and directions, are logical objects sincethey are identified with equivalence classes of logical objects (either concepts or theirextensions), while directions and shapes correspond to equivalence classes of non-logicalobjects (lines and geometrical regions respectively).

11Note that Frege does not seem to be giving a general account of the concept geomet-rical shape, but is instead providing a definition of the narrower concept shape of atriangle. Having noted this, however, we shall from here on ignore it since it is irrelevantto our present concerns.

4

So, the direction of a line λ is identified with the equivalence class of lines100

parallel to λ:101

dir(λ) = –ε(ε||λ)and the shape of a triangle τ is identified with the equivalence class of102

triangles similar to τ :103

shp(τ) = –ε(ε ∼ τ)Step 4: Prove the relevant abstraction principle:104

(∀φ1)(∀φ2)[@C(φ1) = @C(φ2)↔ ΦC(φ1, φ2)]

where:105

@C(φ) = –ε(ΨC(ε, φ))

Thus, given our definition identifying directions with equivalence classes of106

lines, we prove the adequacy of our definition of direction by proving:107

(∀λ1)(∀λ2)[dir(λ1) = dir(λ2)↔ (λ1||λ2)]

that is:108

(∀λ1)(∀λ2)[–ε(ε||λ1) = –α(α||λ2)↔ (λ1||λ2)]and we prove the adequacy of our definition of shape by proving:109

(∀τ1)(∀τ2)[shp(τ1) = shp(τ2)↔ (τ1 ∼ τ2)]that is:110

(∀τ1)(∀τ2)[–ε(ε ∼ τ1) = –α(α ∼ τ2)↔ (τ1 ∼ τ2)]It is important to note that it is Step 3 that provides the definition. Step 4111

amounts to proving that the given definition adequately captures the concept112

being defined: it functions as an adequacy constraint on the definition.113

The examples just discussed are somewhat special since we have here114

applied the simple recipe only to first-order abstractions—that is, to defini-115

tions of mathematical concepts C where the underlying ΦC ’s are objects and116

not second-(or higher-) order concepts, relations, or functions. The reason117

for this is that there is an apparent ambiguity in Frege’s application of this118

construction to concepts, such as the concept cardinal number, whose119

underlying concept ΦC is not objectual. We discuss this in the following120

section in more detail.121

Another aspect in which the definitions of directions and shapes differ122

from the definition of cardinal numbers is that Frege does not explicitly carry123

out Step 4 of the simple recipe for directions or shapes, while he does so124

for cardinal numbers.12 After providing the definition in §68, and before125

sketching the derivation of a version of the Peano axioms in §74-§83, Frege126

has this to say:127

12Frege does, however, motivate these definitions of shape and direction by appeal totheir corresponding abstraction principles—see Grundlagen, §68. But unlike the case ofcardinal numbers, he does not derive them.

5

We will first show that the cardinal number which belongs to the128

concept F is equal to the cardinal number which belongs to the129

concept G if the concept F is equinumerous to the concept G.13130

After sketching a proof of this claim—essentially, the right-to-left direction of131

Hume’s Principle—Frege concludes the section with the following footnote:132

And likewise of the converse: If the number which belongs to the133

concept F is the same as that which belongs to the concept G,134

then the concept F is equal to the concept G.14,15135

Strictly speaking, then, Frege does not provide a full proof sketch of Hume’s136

Principle in §73 of Grundlagen, but that he considers both directions in one137

section we regard as sufficient for our purposes. It is also noteworthy that the138

sections in which Frege sketches both a proof of (the two sides of) Hume’s139

Principle and proofs of central principles of Peano Arithmetic fall under140

the heading “Our definition completed and its worth proved”. Since these141

sections contain a derivation sketch of Hume’s Principle first, and then show142

how to derive the more familiar arithmetic results from Hume’s Principle, it143

seems natural to interpret the derivation of Hume’s Principle as completing144

the definition and thus fulfil Step 4 of Frege’s definitional strategy, while the145

derivation of the Peano axioms demonstrate the worth of the definition.146

To summarise: in Grundlagen Frege provides two sorts of evidence that147

the definition of cardinal numbers as extensions is correct. He sketches a148

proof that the second-order Peano axioms follow from the definition (a task149

carried out with more rigor and in more detail in Grundgesetze). Yet, before150

engaging in the proof, he also notes that the definition entails (each direction151

of) Hume’s Principle. In short, Frege carries out Step 4 of the simple recipe152

when applied to cardinal number and he thus regards Hume’s Principle as153

providing a precise adequacy condition that any definition of the concept154

cardinal number must meet. This, in turn, explains the central role that155

13Frege, Grundlagen der Arithmetik, p. 85.14Frege, Grundlagen der Arithmetik, p. 86n.15Similarly, in the concluding remarks of Grundlagen, Frege emphasizes the fact that

any adequate definition of number must recapture (that is, prove) the relevant principlegoverning recognition conditions, which in the case of cardinal numbers is Hume’s Principle:

“The possibility to correlate single-valuedly in both directions the objects fallingunder a concept F with the objects falling under the concept G, was recognisedas the content of a recognition-judgement for numbers. Our definition, therefore,had to present this possibility as co-referential (gleichbedeutend) with a number-equation. We here drew on similar cases: the definition of direction based onparallelism, of shape based on similarity.” Frege, Grundlagen der Arithmetik,p. 115.

After rehearsing the reasons for rejecting Hume’s Principle itself as a definition in §107,Frege reminds us in §108 of his proof that the explicit definition of cardinal numbersin terms of extensions meets this criterion—that is, he reminds us of his proof of (theright-to-left direction of) Hume’s Principle.

6

Hume’s Principle plays in Grundlagen despite being rejected as a definition156

proper.157

158

ii. two options for identifying abstracts159

In this section, we will outline two ways of unpacking Frege’s identification160

of each number with ‘the extension of the concept “equinumerous to the161

concept F”’. The first option involves understanding the cardinal number of162

F as the extension of the (second-level) concept holding of those concepts163

equinumerous to F—that is:164

”(F ) = –ε(ε ≈ F )

Note that, on the first option, the extension operator –ε binds a first-level165

concept variable, not an object variable.16166

The second option involves understanding the cardinal number of F as167

the extension of the (first-level) concept holding of the extensions of those168

concepts equinumerous to F—that is:169

”(F ) = –ε((∃Y )(ε = –α(Y (α)) ∧ Y ≈ F ))

Although this ambiguity is cleared up in the formal treatment of arithmetic170

in Grundgesetze, we will consider both proposals suggested by the looser171

presentation in Grundlagen. Such an approach will illustrate that, in applying172

the recipe the choice between the first option and the second option is not173

arbitrary or merely a matter of convenience. Instead, there are principled174

reasons for defining cardinal numbers—and, more generally, all second-175

order abstracts—as extensions of first-level concepts holding of extensions of176

concepts. Thus, there are good reasons for Frege—reasons we believe he was177

aware of—to adopt the second option.178

II.1. The First Option. The first way of understanding Frege’s suggestion179

that the cardinal number of the concept F is the extension of the concept180

equinumerous to the concept F is to identify the number of F with181

the extension of the second-level concept holding of all first-level concepts182

equinumerous to F :183

”(F ) = –ε(ε ≈ F )16Frege utilizes extensions of concepts within Grundlagen, while mobilizing the more

general notion of value-ranges of functions within Grundgesetze. Extensions of concepts,however, are a special kind of value-range: they are the value-ranges of unary concepts,where concepts are functions whose range is the True and the False. Since we shall beshifting frequently between Frege’s Grundlagen definition of cardinal number and hisGrundgesetze definition of the cardinal number, we shall use the Grundgesetze notation“–ε(. . . ε . . . )” throughout, taking care to flag when the broader notion of value-range, ratherthan extension, is at issue.

7

If this were the right way to understand Frege, then we can generalise the184

simple recipe to higher-level concepts. Given any mathematical concept185

C, with associated underlying (second-level) concept ΦC and (second-level)186

equivalence relation ΨC , we can identify the C’s as follows:187

@C(F ) = –ε(ΨC(ε, F ))

where @C is the abstraction operator mapping concepts to C’s.188

In particular, applying the first option to extensions themselves provides:189

–α(F (α)) = –ε((∀y)(F (y)↔ ε(y)))

This substitution will be admissible if the recipe is not only a means for190

identifying ‘new’ objects (or, more carefully: for defining new concepts191

by identifying which of the ‘old’ objects—the extensions—fall under those192

concepts) but it is, more generally, a method for identifying any mathematical193

objects.194

There are, we think, good reasons for interpreting the recipe in the195

broader sense: the definitional strategy adopted by Frege in Grundlagen is196

not merely intended to identify which objects are the cardinal numbers, but197

it is intended to play a more general role in Frege’s logicism. In order to198

gain epistemological access to some objects falling under a mathematical199

concept C, a definition has to provide us with identity conditions for the200

objects falling under C. If the recipe achieves this for mathematical objects201

falling under a mathematical concept C via an identification of the objects202

falling under C with particular extensions (and hence applies to at least203

these extensions), then it should apply to all extensions, including those204

objects that do not fall under one or another mathematical/logical concept205

other than extension itself. Otherwise, the domain of extensions would be206

artificially partitioned into two sub-domains corresponding to distinct means207

for determining identity conditions: those extensions that fall in the range of208

a mathematical concept other than extension to which the recipe applies,209

and those that do not.17210

For this reason, we think that Frege’s recipe should also apply to exten-211

sions themselves.18 In that case, however, the first option reading of the212

simple recipe must be rejected as the proper understanding of Frege’s method213

17The point is not that, on the simple recipe, identity conditions for cardinal numbersare not given in terms of Basic Law V. If cardinal numbers are, in fact, extensions, thenthey can be individuated using Basic Law V just like any other extension. The point isthat the philosophically primary identity criterion for cardinal numbers is equinumerosity,which must then be analyzed in terms of the recipe to reduce it to a relation on relevantextensions.

18Note that we do not need the (implausible) claim that this reading of the simple recipeprovides us with a definition of extensions, but rather the weaker claim that applyingthe simple recipe to value-ranges (which, for Frege, require no definition) results, not in adefinition, but in a truth.

8

for defining mathematical concepts and identifying the corresponding objects.214

The reason is simple: The first option is logically incoherent. If we identify215

the extension of a first-level concept with the extension of a second-level216

concept, then we need some general means for settling such cross-level iden-217

tity statements. According to Basic Law V, however, extensions of concepts218

can only be identical when the concepts in question hold of exactly the219

same thing or things. No first-level concept can hold of anything that any220

second-level concept holds of, since first-level concepts hold of objects and221

second-level concepts hold of first-level concepts. As a result (and assuming222

that we extend the notion of extension to second-level concepts in the first223

place) the only logically possible pair < C1, C2 > where C1 is a first-level224

concept, C2 is a second-level concept, and –ε(C1(ε)) = –ε(C2(ε)) is the case225

where C1 and C2 are both empty concepts (although obviously not the ‘same’226

empty concept, since they are of different levels). As a result, the first option227

is not a live option. In particular, the identity in question:228

–α(F (α)) = –ε((∀y)(F (y)↔ ε(y)))

must, at best, always be false, since the degenerate case where both F229

and (∀y)(F (y) ↔ X(y)) hold of nothing whatsoever is not possible here:230

(∀y)(F (y)↔ X(y)) holds of F .231

Now, once Frege had formulated the logic of Grundgesetze in the required232

detail, he would have, no doubt, realised that the first option does not233

apply to extensions (or value-ranges for that matter). If, as we argued above,234

Frege’s recipe has to apply to all mathematical objects, then the failure of the235

first option can now be interpreted in the wider context of motivating a shift236

from the first to the second option. Thus, in contrast to other interpreters,237

we think that Frege’s adoption of the second option in Grundgesetze is not238

merely a choice based on convenience but it is a well-motivated move to fulfil239

the requirements of his recipe.19240

II.2. The Second Option. The second way of understanding Frege’s241

suggestion that the number of the concept F is the extension of the concept242

equinumerous to the concept F is to identify the number of F with the243

extension of the first-level concept holding of the extensions of all first-level244

concepts equinumerous to F :245

”(F ) = –ε((∃Y )(ε = –α(Y (α)) ∧ Y ≈ F )

In order to simplify our presentation, we will now incorporate one of246

Frege’s own tricks: Frege does not define equinumerosity as a second-level247

19Consider, for example, Patricia A. Blanchette Frege’s Conception of Logic (Oxford:Oxford University Press, 2012), p. 83, who interprets Frege’s move from the first to thesecond option as “simply...one of technical convenience”. We pick up on the issue ofarbitrariness in section IV.1. Further discussion of Blanchette’s interpretation of this issuecan be found in Roy T. Cook “Book Symposium: Frege’s Conception of Logic. PatriciaA. Blanchette,” Journal for the History of Analytical Philosophy, iii, 7 (2015): 1-8.

9

relation holding of pairs of first-level relations, but instead defines it as a248

first-level relation holding of the extensions of first-level concepts. Hence,249

“α ≈ β” is true if and only if α and β are the extensions of (first-level) concepts250

Fα and Fβ such that the Fαs are equinumerous to the Fβs.20251

With this new understanding of “≈” in place, Frege’s definition of cardinalnumbers becomes:

”(F ) = –ε(ε ≈ –α(F (α))

This is, essentially, the definition of number provided by Frege in Grundge-252

setze.21253

As was the case with the first option, Frege’s application of the second254

option version of the simple recipe to the concept cardinal number can255

be straightforwardly generalized so as to be applicable to abstracts of any256

first-level concepts. Given any mathematical concept C, with associated257

underlying (second-level) concept ΦC and (second-level) equivalence relation258

ΨC , we can identify the C’s as follows:259

@C(F ) = –ε((∃Y )(ε = §(Y ) ∧ΨC(Y, F )))

where @C is the abstraction operator mapping concepts to C’s. Importantly,260

the second option version of the simple recipe does not result in logical261

incoherence and it is thus an improvement on the first option.262

Nonetheless, the simple recipe does have its limitations: first, the fact263

that the second option depends on identifying abstract objects via equiva-264

lence relations restricts its applicability to unary abstracts, and hence does265

not apply to concepts C where the abstracts falling under C result from266

20Strictly speaking, Frege does not explicitly define equinumerosity in Grundgesetze,but instead defines a ‘mapping into’ operation. His official definition of number involvesa complicated formula involving a complex subcomponent equivalent to equinumerosity,constructed in terms of the ‘mapping into’ construct. The lack of an explicit definitionof equinumerosity in Grundgesetze further emphasizes a fact that we will bring out later:that Hume’s Principle plays no role in the formal development of Grundgesetze.

21There is a difference between the definition of cardinal number that results fromthis reading of the simple recipe and the superficially similar formal definition given inGrundgesetze: Within Grundgesetze Frege’s definition of equinumerosity implies that twofunctions F1 and F2 are equinumerous if and only if there is a one-one onto mappingbetween the arguments that F1 maps to the True and the arguments that F2 maps to theTrue, regardless of whether these functions map all other arguments to the False (thatis, regardless of whether these functions are concepts). Thus, the mature Grundgesetzedefinition of cardinal number does not identify numbers with ‘collections’ of extensions ofequinumerous concepts, but rather with ‘collections’ of value-ranges of functions (includingbut not restricted to concepts) that map equinumerous collections of objects to the True.Along similar lines, the ordered pair of α and β, which shall be examined in detail below,is the ‘collection’ of (the double value-ranges of) all functions that map α and β (in thatorder) to the True, and not the (less-inclusive) ÔcollectionÕ of (value-ranges of) relationsthat relate α to β. This observation, while important for other reasons, is orthogonal toour concerns in this paper, so we ignore it. For further discussion of the issue, see RoyT. Cook “Frege’s Conception of Logic, Patrica A. Blanchette,” Philosophia Mathematica,xxii, 1 (February 2014): 108-20.

10

abstracting off more than one of the underlying ΦC ’s—a problem that would,267

of course, also affect the first option. So, for example, the simple recipe will268

not provide us with a pairing operation (more on this below).269

Second, and more importantly at this stage, the simple recipe gets identity270

conditions wrong in specific cases. And here, once again, the problem is to271

apply the recipe to extensions. Now, while the second option of applying272

the simple recipe does not result in a logical incoherence, we do, however,273

face what we call the problem of the singleton. This problem arises when274

we apply the second option understanding of the simple recipe to extensions275

themselves, obtaining:276

–ε(F (ε)) = –α((∃X)(α = –ε(X(ε)) ∧ (∀y)(F (y)↔ X(y)))

This is equivalent (modulo Basic Law V) to:277

–ε(F (ε)) = –α(α = –ε(F (ε)))

In short, applying this variant of the simple recipe to extensions themselves278

entails (using slightly anachronistic terminology) that every extension is iden-279

tical to its singleton. This result, however, is problematic. If we instantiate280

the formula above with the empty concept C∅:281

–ε(C∅(ε)) = –α(α = –ε(C∅(ε)))

Basic Law V entails that the empty extension –ε(C∅(ε)) and any singleton282

extension are individuated extensionally, hence:283

(∀x)(C∅(x)↔ x = –ε(C∅(ε)))

Since the empty concept C∅ holds of no object, we obtain:284

(∀x)(x 6= –ε(C∅(ε)))

and hence the contradiction:285

–ε(C∅(ε)) 6= –ε(C∅(ε))

It is worth emphasizing that the problem of the singleton does not depend286

in any way on the paradoxical character of Basic Law V itself. So long as287

we accept that the empty extension exists, that the simple recipe applies to288

extensions, and that identity conditions for abstracts are governed by some289

abstraction principle that settle the identity of the empty extension and of290

singletons in terms of co-extensionality (even if it disagrees with Basic Law291

V elsewhere), then the problem will arise.22292

22In particular, the problem of the singleton would still be a problem in formal systemsthat replace the inconsistent Basic Law V with any of the consistent restricted versionsof it explored in recent neo-logicist literature, such as NewV in George Boolos “IterationAgain,” Philosophical Topics, xvii, 2 (Fall 1989): 5-21.

11

So to summarise: while the second option is, in some way, an improvement293

on the first option for identifying abstracts using the simple recipe, it also294

fails as a general recipe for providing identity conditions for all mathematical295

objects. It fails in its application to extensions themselves and it does not296

easily generalise to non-unary abstracts. All this suggests that the simple297

recipe itself is in need of some revisions so to be better-suited for the purposes298

of Frege’s logicism as defended in Grundgesetze. In section 4, we will show299

that the definitions Frege gives in Grundgesetze follow a modified generalized300

recipe.301

Before outlining the new recipe, however, we should ask whether there302

are good reasons for thinking that Frege was aware of the problems affecting303

his simple recipe, and whether there are good grounds for thinking that304

he rejected it for the reasons we have offered. In the next section, we will305

show that Frege was familiar with a version of the problem of the singleton306

by the time of Grundgesetze. This, in turn, provides some evidence that307

his shift from the simple recipe of Grundlagen to the generalized recipe of308

Grundgesetze was quite possibly motivated, in part, by the kinds of concerns309

we have discussed above.310

311

iii. frege on singletons312

The most straightforward explanation of the problem of the singleton is that313

it is brought about by the commitment to identifying extensions and their314

singletons—a commitment implicitly codified in the simple recipe. Identifying315

objects with their singletons generally is implausible at best.23 Frege himself316

was aware of the danger of such an identification and discusses it near the317

end of §10 of Grundgesetze, vol. I.318

Given that in Grundgesetze sentences are names of truth-values, the logic319

of Grundgesetze involves, at a glance, reference to two distinct types of logical320

object: truth-values and value-ranges. In order to reduce the number of types321

of logical objects—with a view to settling all identities within Grundgesetze322

in terms of identity conditions for value-ranges as codified in Basic Law323

V—Frege makes two stipulations. First, he stipulates that the reference324

of true sentences—the True—is to be identified with the extension of any325

concept that holds of exactly the True. In short, the True is identical to the326

singleton of the True:327

The True = (∀x)(x = x) = –ε(ε = (∀x)(x = x))23This is true even though identifying Urelemente—that is, non-sets—with their single-

tons is often convenient and sometimes desirable.

12

Second, he stipulates that the False is identical to the the singleton of the328

False:329

The False = (∀x)(x 6= x) = –ε(ε = (∀x)(x 6= x))

He follows up this observation with the following (rather hefty) footnote:330

It suggests itself to generalise our stipulation so that every object331

is conceived as a value-range, namely, as the extension of a concept332

under which it falls as the only object. A concept under which333

only the object ∆ falls is ∆ = ξ. We attempt the stipulation: let334–ε(∆ = ε) be the same as ∆. Such a stipulation is possible for335

every object that is given to us independently of value-ranges, for336

the same reason that we have seen for truth-values. But before337

we may generalise this stipulation, the question arises whether it338

is not in contradiction with our criterion for recognising value-339

ranges if we take an object for ∆ which is already given to us as340

a value-range. It is out of the question to allow it to hold only for341

such objects which are not given to us as value-ranges, because342

the way an object is given must not be regarded as its immutable343

property, since the same object can be given in different ways.344

Thus, if we insert ‘–αΦ(α)’ for ‘∆’ we obtain345

‘–ε(–αΦ(α) = ε) = –αΦ(α)’346

and this would be co-referential with347

‘ a (–αΦ(α) = a) = Φ(a)’,348

which, however, only refers to the True, if Φ(ξ) is a concept under349

which only a single object falls, namely –αΦ(α). Since this is not350

necessary, our stipulation cannot be upheld in its generality.351

The equation ‘–ε(∆ = ε) = ∆’ with which we attempted this352

stipulation, is a special case of ‘–εΩ(ε,∆) = ∆’, and one can ask353

how the function Ω(ξ, ζ) would have to be constituted, so that354

it could generally be specified that ∆ be the same as –εΩ(ε,∆).355

Then356–εΩ(ε, –αΦ(α)) = –αΦ(α)357

also has to be the True, and thus also358

a Ω(a, –αΦ(α)) = Φ(a),359

no matter what function Φ(ξ) might be. We shall later be ac-360

quainted with a function having this property in ξSζ; however we361

shall define it with the aid of the value-range, so that it cannot362

be of use for us here.24363

24Frege, Grundgesetze der Arithmetik, vol. I., §10, p. 18.

13

There are a few things worth noting regarding this passage. First, Frege364

is clearly aware that we cannot in general identify extensions with their365

singletons, noting that doing so results in identities of the form:366

–ε(F (ε)) = –α(α = –ε(F (ε)))

Such identities can only hold when the concept F holds of exactly one object,367

since this formula entails that F holds of exactly –ε(F (ε)). This is, in essence,368

the same point made above: we assumed that F held of no objects, and369

then derived a contradiction. A similar reductio ad absurdum can of course370

be performed if we assume that F holds of more than one object (and we371

assume that both the extension of F and singletons are individuated in terms372

of co-extensionality).373

Crucially, Frege does more than just point out that the identification of374

extensions with their singletons fails. In addition, he asks whether there is a375

relation R such that:376

–ε(F (ε)) = –α(R(α, –ε(F (ε))))

does, in fact, hold generally. As we have already seen, identity is not such a377

relation, but it is open—as of §10 of Grundgesetze—whether there is some378

other relation R such that the extension of a concept F is identical to the379

extension of the concept “is R-related to –α(F (α))”. For any such R, it must380

be the case that:381

(∀x)(F (x)↔ R(x, –ε(F (ε))))

holds. He then points out that his application operator “S”, which we shall382

return to below, satisfies this constraint.383

What is obvious from all of this is that Frege has a deep understanding384

of the perils that came with identifying extensions with their singletons. But,385

no one was (or likely is) more knowledgable about the technical intricacies386

of the formal system of Grundgesetze and their philosophical implications387

than Frege. Hence, it seems very unlikely that he would not have realized388

the consequences the problem of the singleton has for his earlier definitional389

strategy by the time of Grundgesetze.390

391

iv. identifying abstracts in grundgesetze392

Frege’s Grundlagen definitions (that is, the results of applying the simple393

recipe using the second option), as well as almost all of Frege’s Grundgesetze394

definitions, can be seen as instances of a more general method: the generalized395

recipe. With a single exception—the definition of the application operator396

14

“S”, which we will return to later—Frege’s definitions in Grundgesetze fall397

into three categories:398

First, there are definitions of particular singular terms, such as zero399

“0” (definition Θ), one “1” (definition I), Endlos “i” (definition M), and400

definitions of particular relation symbols such as the successor relation s401

(definition H). With respect to the latter (and other particular relations402

defined later in Grundgesetze), Frege does not provide a definition of the403

successor relation in the modern sense, but rather identifies the object that404

is the double value-range of the relation in question. Hence, these definitions405

are all straightforward identifications of specific objects—in particular, of406

specific single or double value-ranges.407

Second, there are definitions of what modern readers would naturally408

think of as (open or ‘unsaturated’) function or relation symbols, but which409

Frege formalized as (double value-ranges of) functions from value-ranges to410

value-ranges. In addition to the cardinal number operation discussed above411

(a function from concepts to extensions), these include basic operations on412

relations, including the composition of relations p and q (Definition B):413

–α–ε(

r εS(rSp)rS(αSq)

)= pLq

the converse of a relation p (Definition E):414

–α–ε(αS(εSp)) = Up

and the coupling of relations p and q (Definition O):415

–α–ε

a o d c cS(oSp)ε = c;ddS(aSq)α = o;a

= pPq

Each of these definitions identifies a function that takes objects (including416

double value-ranges of relations) as arguments, and provides the double417

value-range of another relation as value.418

Third, we have definitions of what modern readers would naturally identify419

as predicates, but which Frege takes to be function symbols designating420

functions from objects (again, including single or double value-ranges) to421

truth-values. The first example of such a ‘predicate’ is Frege’s definition Γ:422 e d a d = aeS(aSp)eS(dSp)

= Ip

—the definition of the single-valuedness of a relation. Given any particular423

15

double value-range p as argument, this expression denotes a truth-value:25424

the True if the relation is single-valued—that is, if it is a function—and the425

False if it is not.426

So how does Frege arrive at these particular definitions, and why do they427

fall into these three categories? These definitions follow from an application428

of the generalized recipe, which can be rationally reconstructed as follows:429

Step 1: Identify the underlying concept ΦC such that C’s are C’s430

of ΦC ’s.26431

Step 2: Formulate the identity conditions for C’s in terms of432

some appropriate relation ΨC on the underlying domain of ΦC ’s433

such that:434

∀φ1, ...φn, φn+1, ...φ2n ∈ ΦC

[C(φ1, ...φn) = C(φn+1, ...φ2n)↔ ΨC(φ1, ...φn, φn+1, ...φ2n)]

Step 2.5: Via applications of Basic Law V, transform the right-435

hand-side of the biconditional into an identity:436

ΨC(φ1, ...φn, φn+1, ...φ2n)↔ fC(φ1, ...φn) = fC(φn+1, ...φ2n)

Step 3: Identify the C’s with the range of fC . In particular:437

C(φ1, ...φn) = fC(φ1, ...φn)25Of course, as we have already seen, Frege in Grundgesetze identifies truth-values

with value-ranges—in particular, with their own singletons. Thus, Frege’s definitions of‘predicates’ such as “I” are, in fact, functions from value-ranges to value-ranges. SinceFrege’s identification of truth-values with their singletons is never codified in an official basiclaw, however, but occurs instead in ‘unofficial’ philosophical discussion of the formalism(see Grundgesetze §10, vol. I.), the wording given above is preferred.

The point—that is, the real distinction between Frege’s treatment of operations onrelations such as “L”, “U”, and “P”, and his treatment of ‘predicates’ such as “I”—is thathe did not define the latter as the extension of the concept holding of exactly those objectssatisfying the predicate. In short, he did not define “I” as:

–ε(

e d a d = aeS(aSε)eS(dSε)

)= I

parallel to his definitions of “L”, “U”, and “P”, and then write “pSI” instead of “Ip”. Thefact that Frege’s definitions of ‘binary operators’ such as “L”, “U”, and “P” are formulatedas functions from pairs of objects to double value-ranges (which are not truth-values, evenon Frege’s identification of truth-values with their singletons), while unary function symbols(that is, ‘predicates’) such as “I” are defined as functions from objects to truth-values,deserves further scrutiny.

26By the time of Grundgesetze, for Frege the underlying ΦCs are always some classof objects, including single- and double value-ranges. Thus, numbers are, at least in atechnical sense, not directly numbers of first-level concepts, but are instead numbers of theextensions of first-level concepts.

16

The generalized recipe involves two modifications to the simple recipe. The438

first is replacing Step 2 in the former with a more general and flexible439

two-step process (Step 2 and Step 2.5). The second modification is the440

deletion of Step 4. These modifications are both natural and necessary in441

order to carry out the work Frege wishes to carry out in Grundgesetze. In442

the following, we will illustrate how the first modification is essential in443

capturing Frege’s Grundgesetze definitions and explore some technical and444

philosophical consequences of this fact. Then, after three short digressions,445

we will conclude the paper by examining the second modification to the446

recipe, namely abolishing Step 4.447

When applying the generalized recipe to unary operations that provide448

us access to numbers, directions, and shapes, the result is equivalent to that449

obtained when applying the simple recipe, although the details involved in450

getting to this final result are sometimes different.27 For example, letting our451

concept C be cardinal number, the underlying ΦC is just the first-level452

concept extension of a first-level concept. At Step 2 we note that453

cardinal numbers are individuated in terms of equinumerousity—that is:454

(∀φ1)(∀φ2)[”(φ1) = ”(φ2)↔ φ1 ≈ φ2]

Note that the number operator ” now attaches, not to concepts, but to455

objects—that is, φ1 and φ2 are now first-order variables (further, we are again456

utilizing Frege’s understanding of equinumerousity as a relation between457

extensions of concepts).28 We then note that the right-hand side is equivalent458

to:459

(∀z)(z ≈ φ1 ↔ z ≈ φ2)

which, via Basic Law V (and some straightforward logical manipulation) is460

equivalent to:461–α(α ≈ φ1) = –α(α ≈ φ2)

Hence, on the generalized recipe, the cardinal number of x, for any object x,462

is the equivalence class of extensions of concepts equinumerous to x:463

”(x) = –α(α ≈ x)

If x is the extension of a concept:464

x = –ε(F (ε))

however, then on the Grundgesetze account ”(x) will (speaking a bit loosely)465

pick out the same extension as ”(F ) picked out on the Grundlagen simple466

recipe account.467

27In particular, and unlike the simple recipe, on the generalized recipe all definitions willtake objects—usually but not necessarily extensions of concepts—as arguments.

28Note that, as a result, ” is defined for all objects, but it need only be ‘well-behaved’in the intended case, where “x” is the extension of a concept.

17

We can also apply the generalized recipe to arrive at Frege’s definition468

of the pairing operation “;” (Definition Ξ). The concept C in question469

is (ordered) pair. The underlying concept ΦC such that C’s are C’s of470

ΦC ’s is the concept object (Step 1). Things get a bit trickier at Step 2471

and Step 2.5, however, since we are no longer looking for an equivalence472

relation on objects, but an ‘equivalence relation’-like four-place relation. For473

the contemporary reader, with a century of sophisticated set theory under474

her belt, the appropriate relation with which to begin is obvious—pairwise475

identity:476

∀φ1, φ2, φ3, φ4 ∈ ΦC [φ1;φ2 = φ3;φ4 ↔ (φ1 = φ3 ∧ φ2 = φ4)]

We now note that the right-hand-side is equivalent to:29477

(∀R)(R(φ1, φ2)↔ R(φ3, φ4))

which is in turn equivalent to:30478

∀R(φ1S(φ2S–ε–α(Rεα)) = φ3S(φ4S

–ε–α(Rεα)))

which, again via Basic Law V, becomes:479

–ε(φ1S(φ2Sε)) = –ε(φ3S(φ4Sε))

We now have the required identity, and can apply Step 3:480

x; y = –ε(xS(ySε))

and we arrive at Frege’s definition of ordered pair.31481

In order to justify our claim that all of Frege’s Grundgesetze definitions482

(with the exception of “S”) flow naturally from the generalized recipe, it483

is worth working though a different example—the definition of the single-484

valuedness of a function (Definition Γ). This function maps double value-485

ranges of relations to truth-values, so the underlying concept ΦC is just486

double value-range. Equally straightforward is the application of Step487

2—formulating the identity conditions. Since “I” is the sign of a function488

from objects to truth-values, determining the identity conditions for I just489

29We think it is worth noting that it seems likely to us that Frege himself started withthis universally quantified second-order formula.

30We use Frege’s application operator “S” in order to capture Frege’s official definition.We will say more about this operator below, for the moment it can be understood akin tomembership.

31The remainder of Frege’s Grundgesetze definitions, including definitions ∆, K, Λ, N, ΠP, Σ, T, Υ, Φ, X, Ψ, Ω, AA, AB, AΓ follow a similar pattern. In future work we plan toshow explicitly that all of these definitions result from straightforward application of thegeneralized recipe.

18

amounts to determining which arguments are mapped to the True, and which490

arguments are mapped to the False. Hence:491

(∀φ1)(∀φ2)[I(φ1) = I(φ2)↔ ((∀z)(∀w)(zS(wSφ1)→ (∀v)(zS(vSφ1)→ w = v))↔ (∀z)(∀w)(zS(wSφ2)→ (∀v)(zS(vSφ2)→ w = v))]

In short, the truth-value denoted by Iφ1 is identical to the truth-value492

denoted by Iφ2, if and only if the claim that φ1 is the double value-range of493

a single-valued relation (that is, a function) is equivalent to the claim that494

φ2 is the double value-range of a single-valued relation. While this formula495

is complex, we can easily apply Step 2.5 by reminding ourselves that there496

is no distinction between logical equivalence and identity in Grundgesetze.497

Hence the right-hand-side of the above is equivalent to:498

((∀z)(∀w)(zS(wSφ1)→ (∀v)(zS(vSφ1)→ w = v))= (∀z)(∀w)(zS(wSφ2)→ (∀v)(zS(vSφ2)→ w = v))]

and we can now apply Step 3 to arrive at Frege’s definition:499

Ix = (∀z)(∀w)(zS(wSx)→ (∀v)(zS(vSx)→ w = v))

We hope these examples suffice to show that Frege’s Grundgesetze defi-500

nitions share a certain structure which is characterised by the generalized501

recipe. What best explains the (surprising) uniformity of the Grundgesetze502

definitions is that Frege was aware of this recipe—or, at least, one that is very503

much like it— and so, we believe there are good grounds for thinking that504

Frege followed the generalized recipe when composing his magnum opus.32505

Before concluding with a discussion of the consequences of this general506

definitional strategy for an adequate interpretation of Frege’s mature philos-507

ophy of mathematics, there are three issues that need to be addressed: the508

first involves extant criticisms of Frege’s definitions in Grundgesetze to the509

effect that his methodology is completely arbitrary. The second issue is to510

demonstrate that the generalized recipe can be applied in such a way as to511

avoid the problems that plagued the simple recipe—in particular, the problem512

32Admittedly, there is, as far as we know, no explicit mention of a recipe of this kind inFrege’s published writing. According to Albert Veraart “Geschichte des wissenschaftlichenNachlasses Gottlob Freges und seiner Edition. Mit einem Katalog des ursprunglichenBestands der nachgelassenen Schriften Freges,” in Matthias Schirn, ed., Studien zu Frege.3 vols. (Stuttgart–Bad Cannstatt: Friedrich Frommann Verlag, 1976), pp. 49-106, Frege’sNachlaß contained many pages of “formulae” which could have offered us a better insightinto how Frege arrived at the definition that he actually gives. As is well-known, however,most of the Nachlaß was lost during an air raid at the end of the second world war.Compare, however, K. F. Wehmeier and H.-C. Schmidt am Busch,“The Quest for Frege’sNachlass,” in M. Beaney and E. Reck, eds., Critical Assessments of Leading Philosophers:Gottlob Frege (London: Routledge, 2005), pp. 54-68.

19

of the singleton. The final issue is to examine closely the one exception to513

the generalized recipe, in order to show why this case must have been an514

exception on Frege’s account.515

IV. 1. The Generalized Recipe and Arbitrariness. Richard Heck (following516

Michael Dummett), has suggested that Frege’s definitions in Grundgesetze517

are almost entirely arbitrary—that is, that Frege could have chosen just518

about any extensions whatsoever to be the referents of the various notions519

given explicit definitions in Grundgesetze:520

In Frege: Philosophy of Mathematics, Michael Dummett argues521

that Frege’s explicit definition of numerical terms is intended to522

serve just two purposes: To solve the Caesar problem, that is,523

to “fix the reference of each numerical term uniquely”, and “to524

yield” HP (Dummett 1991, ch. 14). The explicit definition is in525

certain respects arbitrary, since numbers may be identified with a526

variety of different extensions (or sets, or possibly objects of still527

other sorts): there is, for example, no particular reason that the528

number six must be identified with the extension of the concept529

“is a concept under which six objects fall”; it could be identified530

with the extension of the concept “is a concept under which only531

the numbers zero through five fall” or that of “is a concept under532

which no more than six objects fall”.33533

However, in a postscript added to this essay in the excellent collection entitled534

Frege’s Theorem, Heck revises his earlier claims. He writes of the arbitrariness535

charge:536

This claim now seems to me to be over-stated, [. . . ]. In particular,537

it now seems to me that there is a strong case to be made that538

the particular explicit definition that Frege gives—assuming that539

we are going to give an explicit definition—is almost completely540

forced. [. . . ]541

So consider the matter quite generally. We have some equivalence542

relation ξRη, and we want to define a function ρ(ξ) in such a543

way as to validate the corresponding abstraction principle:544

ρ(x) = ρ(y)↔ xRy

How, in general, can we do this? So far as I can see, the only545

general strategy that is available here is essentially the one Frege546

adopts: Take ρ(x) to be x’s equivalence class under R, that is,547

the extension of the concept xRξ.34548

33Heck, Frege’s Theorem, p. 95.34Heck, Frege’s Theorem, p. 109.

20

Heck would be absolutely right had Frege applied the simple recipe. In fact,549

the simple recipe, as we described it above, delivers exactly this result!550

As we have already seen, however, the simple recipe is inadequate: it551

is susceptible to the problem of the singleton, and it does not generalize552

straightforwardly to non-unary abstracts. By the time of Grundgesetze,553

Frege had adopted the more powerful, but also more flexible, generalized554

recipe. As a result, the correct reading of Frege’s mature Grundgesetze555

definitions is somewhere between the ‘completely arbitrary’ understanding556

suggested by the initial Heck quote and the ‘completely forced’ understanding557

suggested by the postscript. In fact, any of the definitions Heck considers558

in the passage above could be arrived at via the generalized recipe as ‘the’559

definition of cardinal numbers.560

Since constructions of such alternate definitions of cardinal numbers561

are familiar, we will illustrate this phenomenon with a different example—562

Frege’s definition of the ordered pair operation “;”. Recall that we began563

our reconstruction of Frege’s definition of ordered pairs (as, loosely speaking,564

sets of all relations that relate the objects in question in the appropriate565

order) by noting that the following provides the correct identity conditions.566

∀φ1, φ2, φ3, φ4 ∈ ΦC [φ1;φ2 = φ3;φ4 ↔ (φ1 = φ3 ∧ φ2 = φ4)]

Thus, Step 1 and Step 2 remain as before. The difference comes in how we567

carry out Step 2.5. Here, we will note that the right-hand side of the above568

is equivalent to:569

(∀x)(∀y)((x = φ1 ∧ y = φ2)↔ (x = φ3 ∧ y = φ4))

Two applications of Basic Law V then provide:570

–α–ε(ε = φ1 ∧ α = φ2) = –α–ε(ε = φ3 ∧ α = φ4)

We now have the required identity, and can apply Step 3, resulting in the571

following definition of ordered pair:572

x; y = –α–ε(ε = x ∧ α = y)

In short, on this application of the generalized recipe, the ordered pair of x573

and y is not (speaking loosely) the set of all relations that holds of x and y574

(in that order), but is instead the single relation that relates x to y (again,575

in that order) and relates nothing else to anything else.35576

35Other paths to the requisite identity on the right hand side are possible. For example,one can carry out Step 2.5 in such a way as to arrive at a Fregean version of the Kuratowskidefinition of ordered pair—that is:

x; y = –ε(ε = –α(α = x) ∨ ε = –α(α = x ∨ α = y))

Details are left to the reader.

21

Thus, the generalized recipe does not generate a unique definition, but577

it is instead a general method for arriving at one of a number of equally578

adequate definitions. As a result, there is a measure of arbitrariness present579

in Frege’s mature account of definitions in Grundgesetze. This point should580

not be overstated, however: The method does not license an ‘anything-goes’581

approach to definition. In particular, it is not the case that given any582

acceptable definition of the form:583

f(x1, x2, ...xn) = Φ(x1, x2, ...xn)

and any arbitrary one-to-one function g, that:584

f(x1, x2, ...xn) = g(Φ(x1, x2, ...xn))

is also an acceptable definition. On the contrary, according to the generalized585

recipe, any acceptable definition must proceed by moving from appropriate586

identity conditions (Step 2) via logical laws (including Basic Law V) to587

an appropriate identity (Step 2.5). Thus, while the generalized recipe is588

open-ended—sanctioning more than one possible definition but, presumably,589

allowing no more than one at once—it does not sanction just any definition590

that might get the identity conditions correct.591

A final question remains: Why did Frege select the particular definitions592

that he did select, rather than one or another of the other possibilities?593

Here we can at best speculate, but we suspect the answer will lie in a594

combination of two factors. First, there is the issue of technical convenience.595

Some generalized recipe definitions of a particular concept will be more596

fruitful or more economical than others in terms of the role they play in597

the constructions and proofs that Frege wishes to carry out in Grundgesetze.598

Second, there is the issue of applications and what is now called Frege’s599

Constraint—the thought that an account of the application of a mathematical600

concept should flow immediately from the definition of that concept (see,601

Frege, Grundgesetze der Arithmetik, vol. II., p. 157). Clearly, some definitions602

will satisfy Frege’s Constraint more easily and more straightforwardly than603

others.36604

Making such judgements with regard to one proposed definition rather605

than another will not always be simple, however. At an intuitive level,606

both the convenience/fruitfulness/economy consideration and the Frege’s607

Constraint consideration seem to weigh in favor of Frege’s preferred definition608

of number rather than any of the alternative constructions suggested by609

Heck. But the case for Frege’s preferred definition of ordered pair, rather610

than the alternative construction just given, is not so clear. It will require a611

36An obvious third consideration is simplicity. Thus, it would be perverse for Fregeto identify numbers with the singletons of the objects that he does identify as numbers,even though such a definition can be obtained via the generalized recipe and does get theidentity conditions for numbers correct.

22

detailed examination of the role that ordered pairs play in the derivations of612

Grundgesetze and the way the notion of pair is applied more generally. For613

now, since we have other fish to fry, we will remain content having raised614

this interpretational question.37615

IV.2. Arbitrariness and the Problem of the Singleton. Since we began this616

section with the observation that the explicit definitions given in Grundlagen617

can be recaptured by application of the generalized recipe, the natural question618

to ask next is whether an application of the generalized recipe to extensions619

themselves will fall prey to the problem of the singleton. The answer to this620

question is, in an interesting and important sense, “yes” and “no”. In more621

detail: some applications of the generalized recipe do run afoul of the problem622

of the singleton, but not all do.623

In applying the generalized recipe to extensions, Step 1 and Step 2 are624

straightforward: the concept C in question is extension, the underlying625

concept ΦC such that C’s are C’s of ΦC ’s is the concept (first-level)626

concept, and the appropriate equivalence relation on concepts is coexten-627

sionality. Thus, we have:628

–ε(F (ε)) = –ε(G(ε))↔ (∀x)(F (x)↔ G(x))

We can now apply Basic Law V to the right-hand side, and obtain:629

–ε(F (ε)) = –ε(G(ε))↔ –ε(F (ε)) = –ε(G(ε))

With Step 2.5 completed, we can apply Step 3 and obtain the following630

innocuous identity:631–ε(F (ε)) = –ε(F (ε))

So far, so good—the most natural way of applying the generalized recipe632

turns out to be immune to the problem of the singleton.633

The problem is that Step 2.5—the real culprit in the arbitrariness issue—634

only requires that we transform the right-hand side of the abstraction principle635

formulated in Step 2 into an identity. It does not provide any particular636

guidance on how to do so, nor does it guarantee that there will be only one637

such identity that can be reached via the application of basic laws and rules638

of inference. Thus, in carrying out Step 2 above, we could have applied Basic639

Law V to the right-hand side to obtain:640

–ε(F (ε)) = –ε(G(ε))↔ –ε(F (ε)) = –ε(G(ε))

then applied some basic logic to obtain:641

–ε(F (ε)) = –ε(G(ε))↔ (∀x)(x = –ε(F (ε))↔ x = –ε(G(ε)))37Needless to say, we plan to return to this issue in future work.

23

and then applied Basic Law V again to obtain:642

–ε(F (ε)) = –ε(G(ε))↔ –α(α = –ε(F (ε))) = –α(α = –ε(G(ε)))

Applying Step 3 at this stage would result in the following identification:643

–ε(F (ε)) = –α(α = –ε(F (ε)))

This, however, is exactly the identity that got us into trouble in the first place.644

Thus, the generalized recipe can be applied safely to extensions themselves,645

but not all such applications are safe. What, then, does this tell us about646

the arbitrariness of the generalized recipe itself, and how we are meant to647

apply it in potentially problematic cases?648

One possible response is to formulate some additional principles guiding649

the application of the recipe—principles that legitimate the first of the two650

applications of the generalized recipe to extensions, while ruling out the651

second application as illegitimate. Supplementing the recipe in this manner,652

if it were possible, could perhaps be done in such a way as to eliminate all653

arbitrariness whatsoever, salvaging the idea that Frege’s methods provide654

a unique definition of each mathematical concept. Such an account would655

be attractive, but let us raise a problem for it (though there might be many656

more).657

It is not clear how to formulate such constraints on Step 2.5 in the first658

place. In comparing the two constructions above, one is immediately struck659

by the fact that, in the second, problematic construction, we had already660

obtained an identity of the requisite form:661

–ε(F (ε)) = –ε(G(ε))↔ –ε(F (ε)) = –ε(G(ε))

but then continued to manipulate the right-hand side until we had obtained a662

second such identity, to which an application of Step 3 provided the problem663

of the singleton-susceptible definition. Thus, one natural thought is to664

require that the application of Step 2.5 terminate at the first instance of an665

appropriate identity on the right-hand side. While such a rule would block666

the second construction above, it does not block an alternate construction667

that terminates with the same identity, and hence (via application of Step668

3) provides the same problematic identification of extensions with their669

singletons. We begin with the same equivalence relation on the right, and,670

applying some straightforward logic, arrive at:671

–ε(F (ε)) = –ε(G(ε))↔ (∀H)((∀x)(H(x)↔ F (x))↔ (∀x)(H(x)↔ G(x)))

Two applications of Basic Law V provide us with:672

–ε(F (ε)) = –ε(G(ε))↔ (∀H)(–ε(H(ε)) = –ε(F (ε))↔ –ε(H(ε)) = –ε(G(ε)))

24

We then obtain:673

–ε(F (ε)) = –ε(G(ε))↔ (∀x)(x = –ε(F (ε))↔ x = –ε(G(ε)))

via more logic, and apply Basic Law V in order to obtain the problematic674

identity:675

–ε(F (ε)) = –ε(G(ε))↔ –α(α = –ε(F (ε))) = –α(α = –ε(G(ε)))

Thus, requiring that Step 2.5 halts at the first appropriate identity does not676

block the problematic construction.38677

That being said, there obviously is something fishy about the implemen-678

tations of the generalized recipe that results in the problem of the singleton.679

Of course, it is possible that Frege would have rejected these constructions680

based on the sort of consideration discussed in the previous section: they681

introduce an understanding of the concept extension that is less convenient,682

less fruitful, and less simple than the original construction (and maybe they683

also violate Frege’s Constraint). But there is another reason for rejecting684

them as legitimate applications of the generalized recipe: they violate logical685

constraints on the provision of adequate identity conditions for mathematical686

objects. Since the generalized recipe proceeds via explicit consideration of687

such identity conditions, it seems plausible that any application of the recipe688

should, in the end, respect such constraints. Frege was well aware of the689

need to respect logical and metaphysical constraints when proposing identi-690

ties: Frege’s permutation argument in §10 of Grundgesetze is, in effect, an691

argument which shows that identifying the truth values with their singletons692

will not generate logical difficulties of exactly the sort that would arise were693

he to identify all objects with their singletons more generally.39694

This provides an additional criterion by which Frege might judge particu-695

lar applications of the recipe, and which can thus be used to help explain why696

he arrived at the particular definitions codified in Grundgesetze: in addition697

to respecting considerations of simplicity and fruitfulness, and adhering to698

Frege’s Constraint, applications of the generalized recipe should not bring699

with them logical difficulties of the sort exemplified by the problem of the sin-700

gleton.40 From this perspective, then, the fact that there is a well-motivated701

38Note that, although Frege does not distinguish between biconditionals and identities,the intermediate formulas in the construction above involve universal quantifications ofidentities/biconditionals, and hence are not identities themselves.

39Of course, the identification of truth values with their singletons is, as we have alreadyemphasized, not carried out via an official definition or axiom within the formal system ofGrundgesetze, but is instead merely a ‘meta’-level methodological stipulation. Nevertheless,the discussion in §10 of Grundgesetze makes it clear that Frege was explicitly aware of thesort of logical constraints that weigh in favor of the simpler application of the generalizedrecipe to extensions.

40Note that the sort of logical difficulty at issue is not restricted to applications of thegeneralized recipe to extensions, but would also apply if Frege were to codify his identification

25

implementation of the generalized recipe that does not give rise to the problem702

of the singleton, might well be enough to regard the generalized recipe to be703

in good standing with respect to that very problem.704

IV.3. The Exception to the Generalized Recipe. The only exception to the705

generalized recipe is definition A—the definition of the application operation706

“S”:707

K–α(

g g(a) = αu = –εg(ε)

)= aSu

aSu is the value of the function f applied to the argument a where u is708

the value-range of f (when u is not a value-range, then aSu refers to the709

value-range of the function that maps every object to the false—that is, to710–ε( ε = ε).)711

Of particular interest is the case where u is the extension of a concept712

C (that is, C is a function from objects to truth-values), where aSu will713

be the True if C holds of a, and the False otherwise. As a result, when714

applied to extensions, S is, in effect, a Fregean analogue of the set-theoretic715

membership relation ∈, and Frege often uses S as a membership relation on716

extension of concepts.41717

What is most notable about S for our purposes, however, is that it is718

an exception to the account of the Grundgesetze definitions sketched above:719

FregeÕs application operator S is neither a definition of a specific object nor720

is it the result of an application of the generalized recipe to obtain definitions721

of unary predicates or definitions of binary functions on value-ranges, but722

it is a unique fourth case. It is therefore likely no accident that this is the723

very first definition Frege provides in Grundgesetze, since it not only plays a724

critical role in the later constructions (as a quick perusal of its use in the725

remaining definitions and central theorems makes clear), but it also plays a726

unique role in Frege’s approach to definition.727

In order to see why definition A is special, it is worth working through728

what would result if we attempted to arrive at a definition of “S” via the729

generalized recipe. S is a function that takes two objects as arguments, and,730

when the latter argument is the value-range of a function, gives the value of731

that function applied to the first argument. Hence, applying Step 1 and Step732

2 of the recipe, we obtain something like:733

(∀x)(∀y)(∀z)(∀w)[xSy = zSw

↔ (∀v)[(∃f)(y = –ε(f(ε) ∧ f(x) = v)↔ (∃f)(w = –ε(f(ε) ∧ f(z) = v)]

of truth values with their singletons within the formal system of Grundgesetze. Similarlogical constraints would govern cases where the generalized recipe were applied to twodistinct concepts with non-disjoint extensions, since the definitions would need to belogically compatible on those objects falling under both concepts.

41Frege himself glosses this operation as the “Relation of an object falling within theextension of a concept” Frege, Grundgesetze der Arithmetik, vol.I., p. 240.

26

Via Basic Law V, we can see that the right-hand side of the formula above734

is equivalent to:735

–ε((∃f)(y = –ε(f(ε) ∧ f(x) = ε)) = –ε((∃f)(w = –ε(f(ε) ∧ f(z) = ε))

With the identity required by Step 2.5 in hand, we can then suggest the736

following definition:737

xSy = –ε((∃f)(y = –ε(f(ε) ∧ f(x) = ε))

This definition gets the identity conditions right, but there is an immediate,738

and obvious, problem: This definition does not give us the value of the739

function f applied to argument x, where y = –ε(f(ε)), but instead provides740

us with the singleton of f(x). As we have already shown, however, Frege741

was quite aware of the dangers of haphazardly conflating objects with their742

singletons, so it should come as no surprise that Frege does not adopt the743

incorrect definition above, but instead applies the ‘singleton-stripping’42744

operation K to this formulation, obtaining the correct definition:745

xSy = K–ε((∃f)(y = –ε(f(ε) ∧ f(x) = ε))

Thus, Definition A is the sole exception to the generalized recipe since it746

requires an additional step.747

Why is Definition A different from the remaining definitions in Grundge-748

setze? The answer is surprisingly straightforward. Throughout the rest of749

Grundgesetze, each definition introduces a new concept, function, or other750

operation by identifying the range of that concept, function, or operation with751

a sub-collection of the universe of value-ranges. In short, Frege is defining752

new concepts by identifying their ranges with objects taken from the old, and753

constant, domain. As a result, it is sufficient for his purposes in these cases754

merely to identify some objects with the right identity conditions (modulo755

the possible additional constraints touched on in the previous subsections),756

and this is exactly what the generalized recipe accomplishes.757

With the definition of S something very different is going on. In this758

case, Frege is not attempting to introduce some new concept, instead he is759

attempting to formulate a new way of getting at an already understood and760

fully specified operation—function application. As Frege puts it:761

42The ‘singleton stripping’ (or backslash) operator is a unary function from objects toobjects such that (see Frege, Grundgesetze der Arithmetik, vol. I., §11, p. 19):

f(a) = b if a = –ε(b = ε)= a otherwise.

In short, Frege’s backslash is an object-level function that, when applied to the extensionof a concept, serves the same purpose as a Rusellian definite description operator whenapplied directly to that concept.

27

It has already been observed in §25 that first-level functions can762

be used instead of second-level functions in what follows. This763

will now be shown. As was indicated, this is made possible by764

the fact that the functions appearing as arguments of second-765

level functions are represented by their value-ranges, although of766

course not in such a way that they simply concede their places to767

them, for that is impossible. In the first instance, our concern is768

only to designate the value of the function Φ(ξ) for the argument769

∆, that is, Φ(∆), using ‘∆’ and ‘–εΦ(ε)’. 43770

In short, Frege needs a definition of “S” that not only guarantees that the771

objects ‘introduced’ have the right identity conditions, but in addition that772

they are the right objects. As a result, Definition A is of a very different sort773

than the definitions that follow it, and so it should not be surprising that it774

does not follow the pattern provided by the generalized recipe.775

776

v. applications and consequences777

We believe that the general recipe not only provides an accurate and illumi-778

nating rational reconstruction of Frege’s method of definition in Grundgesetze,779

but that he knowingly applied this methodology (or something very similar).780

As mentioned before, it would be hard to explain the uniformity of the781

Grundgesetze definitions if Frege did not have a methodological template782

of this sort in mind. However, we shall not here offer a further defence of783

the claim that Frege’s use of the generalized recipe was explicit. Instead, we784

shall conclude by showing how awareness and appreciation of the role of the785

generalized recipe in Frege’s Grundgesetze can shed a new light on a number786

of difficult interpretative issues in Frege scholarship.44787

V.1. The Role of Basic Law V in Grundgesetze. As has been shown788

by Richard Heck45 Frege did not make much real use of Basic Law V in789

the derivations found in part II of Grundgesetze—Frege’s only ineliminable790

appeal to Basic Law V is in deriving each direction of Hume’s Principle.791

Most other occurrences of value-ranges, and applications of Basic Law V to792

manipulate them, are easily eliminable. This raises a fundamental question793

about the role of Basic Law V in Frege’s philosophy of mathematics—one794

forcefully formulated by Heck:795

43Frege, Grundgesetze der Arithmetik, vol. I., §34, p. 52.44Here we will address only two such topics, but we believe that the account of definition

given here can also provide insights into the Caesar problem, Frege’s reconstruction of realanalysis, and his views on geometry, amongst other things. We plan on returning to thesetopics in future work.

45See Richard G. Heck, Jr. “The Development of Arithmetic in Frege’s Grundgesetze derArithmetik,” in Journal of Symbolic Logic, lviii, 2, (1993): 579-601.

28

How can an axiom which plays such a limited formal role be of796

such fundamental importance to Frege’s philosophy of mathemat-797

ics?46798

Clearly, Frege did attach fundamental importance to Basic Law V. Consider,799

for example, the Afterword of Grundgesetze, where, faced with Russell’s800

paradox, he attempts to provide a ‘correction’ to his conception of value-801

ranges. Frege does not, as might be expected given the limited formal role802

that Basic Law V plays, suggest that we abandon extensions altogether,803

but instead suggests that a slight modification of our understanding of804

value-ranges is all that is needed:805

So presumably nothing remains but to recognise extensions of806

concepts or classes as objects in the full and proper sense of807

the word, but to concede at the same time that the erstwhile808

understanding of the words “extension of a concept” requires809

correction.47810

After he introduces the principle that introduces the ‘improved’ understanding811

of extensions—Basic Law V′—he closes the Afterword by stating that:812

This question may be viewed as the fundamental problem of813

arithmetic: how are we to apprehend logical objects, in particular,814

the numbers? What justifies us to acknowledge numbers as815

objects? Even if this problem is not solved to the extent that I816

thought it was when composing this volume, I do not doubt that817

the path to the solution is found.48818

So, for Frege there is no doubt that something in the spirit of Basic Law V819

captures the “characteristic constitution” of value-ranges, and that value-820

ranges play a central role in his philosophical project—a role they continue821

to play even when confronted with the paradox. But how are we to square822

Frege’s insistence on the importance of Basic Law V (or some variant of823

it such as Basic Law V′) with the limited formal role that it plays in the824

derivations of Grundgesetze?825

Our interpretation of Frege’s Grundgesetze highlights a central role played826

by Basic Law V—one distinct from its role as an axiom within the formal827

system of Grundgesetze. The generalized recipe relies fundamentally on Basic828

Law V (or, more carefully, on a metatheoretic analogue of Basic Law V which829

is first introduced in vol. I., §3 and §9), since applications of Basic Law V830

are required (in most cases) in order to move from the statement of identity831

conditions (Step 2) to the required identity between objects (Step 2.5). Thus,832

the role played by Basic Law V (and, later, by Basic Law V′) is broader833

46Heck, Frege’s Theorem, p. 65.47Frege, Grundgesetze der Arithmetik, vol. II., pp. 255-56 (our italics).48Frege, Grundgesetze der Arithmetik, vol. II., p. 265.

29

than merely providing the identity conditions for value-ranges. Instead, it834

also plays a central role in identifying which value-ranges are the objects835

‘falling under’ all other mathematical concepts. In short, its role is not only836

logical—as a central principle of the formal system of Grundgesetze—but837

also epistemological and metaphysical, since it is a central component of the838

method by which we define mathematical concepts and identify mathematical839

objects such as cardinal numbers and ordered pairs. As a result, and in840

retrospect, it should not be too surprising that Basic Law V plays a limited841

role in the formal proofs of Grundgesetze, since this formal work consists842

merely of unpacking the real work carried out by Basic Law V: the (informal,843

metatheoretical) formulation of accurate and adequate definitions prior to844

formal derivations—that is, its role in the generalized recipe.49845

V.2. The Role of Hume’s Principle in Grundgesetze. A second issue of846

interest here, and extensively discussed in Richard Heck’s writings50 is the847

role of Hume’s Principle in Frege’s mature philosophy of mathematics. As we848

noted in section 1, Frege appeals to Hume’s Principle in §62 of Grundlagen849

when attempting to explain how numbers are given to us. Frege ultimately850

rejects Hume’s Principle as a definition of number and opts instead for the851

explicit definition of cardinal numbers as a type of extension. Nevertheless,852

Frege explicitly requires that any value-range based definition should allow853

us to prove Hume’s Principle, and Hume’s Principle continues to plays a854

central role throughout the remainder of Grundlagen.855

Given the continued appearance of Hume’s Principle (and similar informal856

principles) throughout Grundlagen, we think that this principle played two857

separate (but interrelated) roles in Frege’s philosophy of mathematics at this858

point even after it was rejected as a definition. First, Hume’s Principle as an859

informal meta-theoretical principle provides the correct identity conditions860

for cardinal numbers and guides the formulation of a definition of cardinal861

numbers (that is, whatever extensions are chosen, they must have the identity862

conditions codified by Hume’s Principle). Second, Hume’s Principle, as a863

formula of the—in Grundlagen informal—object language, constitutes an864

adequacy condition on any explicit definition of cardinal numbers in terms of865

extensions (or in terms of anything else, for that matter): whatever definition866

we choose, it must demonstrably provide the right identity conditions; the867

way to provide such a guarantee is to require that it proof-theoretically entails868

49See P. A. Ebert and M. Rossberg “Mathematical Creationism in Frege’s Grundgesetze,”in P. A. Ebert and M. Rossberg, eds., Essays on Frege’s Basic Laws of Arithmetic (Oxford:Oxford University Press, forthcoming), where we argue that Frege draws on exactly thisfurther role of Basic Law V in the rather intriguing passages §146 and §147 of volume II ofGrundgesetze.

50See, for example, Richard G. Heck, Jr. “Frege’s Principle,” in J. Hintikka, ed., FromDedekind to Godel: Essays on the Development of the Foundations of Mathematics (Dor-drecht: Kluwer Academic Publishing, 1995), pp. 119-45, and Richard G. Heck, Jr. “JuliusCaesar and Basic Law V,” in Dialectica, lvix, 2 (2005): 161-78, as well as Richard G. Heck,Jr. Reading Frege’s Grundgesetze (Oxford: Oxford University Press, 2012).

30

the formula that codifies those identity conditions—that is, the definition869

must entail Hume’s Principle.870

This all seems straightforward enough, but we now arrive at a puzzle: why871

is it that Frege does not mention Hume’s Principle, or even explicitly prove it872

in full biconditional form, in Grundgesetze? Frege does prove each direction873

individually, but he does not put them together into a biconditional/identity874

claim. As already noted, Frege does not explicitly prove Hume’s Principle in875

full in Grundlagen either, but the proof sketch of the right-to-left direction876

in §73, plus the footnote at the end of the same section addressing the left-to-877

right direction, are, we think, meant to jointly indicate the existence of such a878

proof. Moreover, Frege often talks of Hume’s Principle in biconditional form879

in the prose in Grundlagen. In contrast, in Grundgesetze the two directions880

of Hume’s Principle are proven in different chapters (A and B respectively)881

with no indication that they are to be ‘put together’ or that anything might882

be gained by doing so.51 Also, there is no mention of Hume’s Principle as883

a biconditional in the prose of Grundgesetze. Interestingely, the sections of884

Grundgesetze where the definition of natural number is provided (§§38-46)885

refer to §68 of Grundlagen (where the explicit definition of cardinal number886

is first given), §§71-72 of Grundlagen (where the definition of equinumerosity887

is formulated), and §§74-79 of Grundlagen (where explicitly definitions of888

0, 1, and successor are formulated, and sketches of the Peano axioms are889

given). Striking in its absence is any mention of §73 of Grundlagen where890

the sketch of the proof of Hume’s Principle is given.52 Taken together, this891

suggests that the role of abstraction principles in Grundgesetze has changed892

and that Hume’s Principle, understood as a constraint on any adequate893

definition of cardinal number, has disappeared in Grundgesetze. How are we894

to reconcile the fundamentality of Hume’s Principle in the philosophy of the895

Grundlagen-Frege with the fact that it plays a far less important role in the896

philosophy of the Grundgesetze-Frege?897

Once we are aware of the difference between the simple recipe and the898

51We owe this important observation to R. May and K. Wehmeier, “The Proof of Hume’sPrinciple,” in P. A. Ebert and M. Rossberg, eds., Essays on Frege’s Basic Laws of Arithmetic(Oxford: Oxford University Press, forthcoming). Although they give a different explanationfor this odd fact than the one given here. They are the first to suggest that Frege’s failureto ‘conjoin’ the two directions of Hume’s Principle is not merely a technical quirk of theorganization of Grundgesetze, but instead provides insights into what Frege was up to.Thus, our own discussion owes much to their careful examination of these issues.

52Frege does indeed mention §73 of Grundlagen later, in a footnote which we reproducein its entirety:

“Compare Grundlagen, p. 86.” Frege, Grundgesetze der Arithmetik, vol. I.,§54, p. 72n.

This footnote does not concern the derivation of Hume’s Principle in Grundlagen §73,however, but merely highlights the fact that Frege’s definition and elucidation of thecomposition relation in Grundgesetze §56 is based on notions first presented in a sub-portion of that derivation.

31

generalized recipe, an explanation is not hard to come by. Sometime between899

Grundlagen and Grundgesetze Frege must have realized that, if Step 2 of the900

generalized recipe is carried out correctly—that is, if, in the case of cardinal901

numbers, Hume’s Principle (or the metatheoretical analogue given above) is902

used to provide the identity conditions for cardinal numbers—then Step 4 of903

the simple recipe is redundant. There simply is no need to proof-theoretically904

establish Hume’s Principle, qua abstraction principle, within the formalism905

of Grundgesetze so long as the generalized recipe is carried out correctly,906

and nothing that is of philosophical or mathematical importance would be907

achieved by putting together both sides of Hume’s Principle and proving the908

formal counterpart in the language of Grundgesetze.909

As a final observation, it is worth noting that these points might also910

help to explain why Frege was not at all tempted to use Hume’s Principle as911

a definition of cardinal number after he became aware of Russell’s paradox,912

especially given Frege was arguably aware of the fact that Hume’s Principle913

alone would entail all of the Peano axioms.53 Dropping Basic Law V leaves914

Frege without a general means for defining mathematical objects—that is,915

it forces him to abandon the generalized recipe (and the simple recipe, for916

that matter) altogether. Hume’s Principle, or its metatheoretical analogue,917

can (and does) provide the right identity conditions for cardinal numbers,918

but it is insufficient to pick out which objects the cardinal numbers are.919

Hume’s Principle simply cannot play the epistemological and metaphysical920

role that Basic Law V was meant to play in Grundgesetze. Thus, without921

Basic Law V (or some variant, such as Basic Law V′) Frege was left with no922

means for defining and thereby introducing mathematical objects, and hence923

no identifiable mathematical objects at all. This observation is, of course,924

in stark contrast to the recent neo-logicist attempt to found arithmetic on925

Hume’s Principle. In the following, we want to highlight one more important926

difference between the two approaches.54927

V.3. The Definitional Strategy and Neo-Logicism. Finally, it is worth ob-928

serving that the ontology presupposed and utilized by Frege in his application929

of the generalized recipe—or even the simple recipe—differs markedly from930

recent neo-logicist approaches as defended by Bob Hale and Crispin Wright55931

53In fact, he does briefly consider, and immediately rejects, this option in a letter to Rus-sell. See G. Frege, “Letter to Russell, XXXVI/7,” in G. Gabriel, H. Hermes, F. Kambartel,C. Thiel, and A. Veraart, eds., Gottlob Frege: Wissenschaftlicher Briefwechsel (Hamburg:Meiner Verlag, 1976), p. 224.

54Compare Patricia Blanchette “The Breadth of the Paradox,” Philosophia Mathematica,xxiv, 1 (February 2016): 30-49, who highlights further differences between the “Scottish”neo-logicist and Frege’s logicism.

55See Crispin Wright Frege’s Conception of Numbers as Objects (Aberdeen: AberdeenUniversity Press, 1983) and B. Hale and C. Wright The Reason’s Proper Study: Essaystowards a Neo-Fregean Philosophy of Mathematics (Oxford: Oxford University Press,2001). For an overview of issues concerning this form of neo-logicism, see P. A. Ebert andM. Rossberg, introduction to Abstractionism (Oxford: Oxford University Press, 2016),

32

Despite their Fregean roots, neo-logicists reject the idea that objects falling932

under some mathematical concept C should be identified with corresponding933

extensions or value-ranges. Instead, given a mathematical concept C, the934

neo-logicist will provide an abstraction principle of the form:935

(∀α)(∀β)[@C(α) = @C(β)↔ EC(α, β)]

that defines the concept C by providing identity conditions (via the equiva-936

lence relation EC(. . . , . . . )) for abstract objects falling under C (the referents937

of abstraction terms @C(. . . )). Hence, on the neo-logicist approach cardinal938

numbers and other abstract objects are not identified as being amongst some939

more inclusive, previously identified range of objects. As result, a neo-logicist940

does not require anything akin to Frege’s recipe and, thus, she is not plagued941

by the sort of limited arbitrariness discussed previously: the abstract objects942

falling under mathematical concepts just are whatever objects are delineated943

by (acceptable) abstraction principles.944

This plenitude of kinds of abstract objects comes at a cost, however:945

the neo-logicist owes us a principled account of the truth-conditions of946

cross-abstraction identity statements of the form:947

@C1(α) = @C2(β)

where C1 and C2 are different mathematical concepts, defined by differ-948

ent abstraction principles. This problem has come to be called the C−R949

problem.56950

In contrast, such cross-abstraction identities are easily resolved by Frege:951

given two mathematical objects whose identity or distinctness might be952

in question, we need merely determine which extensions the generalized953

recipe identifies with those objects, and then apply Basic Law V to settle954

the identity claim in question.57 Of course, given the arbitrariness in the955

generalized recipe, it is possible that two objects that have been defined in956

such a way as to be distinct might have been defined in some other manner957

such that they would have been identical. But once a particular choice is958

pp.1-30.56The name is a play on the familiar phrase “the Caesar problem”, and refers to the

specific case of determining whether the real numbers R generated by one abstractionprinciple are identical to a sub-collection of the complex numbers C given by a distinctabstraction principle. For a fuller discussion of this problem, see R. T. Cook and P. A. Ebert,“Abstraction and Identity,” Dialectica, lvix, 2 (2005): 121-39, and more recently in PaoloMancosu “In Good Company? On Hume’s Principle and the Assignment of Numbers toInfinite Concepts,” Review of Symbolic Logic, viii, 2 (June 2015): 370-410.

57We do not mean to imply that settling whether two extensions in a non-well-foundedtheory of extensions such as that found within Grundgesetze (or consistent sub fragmentsof Grundgesetze is trivial or effective. The point is merely that on Frege’s view the recipeentails that there will be a straightforward fact of the matter that settles these identities.Hence, regardless of whether determination of the status of cross-abstraction identities isin-principle mathematically difficult, there are no deep philosophical puzzles here.

33

made, there is no C−R problem within Frege’s original logicist project as959

developed in Grundgesetze.960

As a result, we can now understand one aspect of the relation between961

Frege’s logicism and his modern day neo-logicist successor in terms of adopt-962

ing different approaches to a particular trade-off: Frege, in adopting the963

generalized recipe, was forced to accept some arbitrariness with regard to how964

he defined mathematical concepts such as cardinal number and ordered965

pair. Once he has settled on particular definitions, however, there are no966

further questions regarding identity claims holding between mathematical967

objects: all such objects are extensions (or value-ranges more generally) and968

so Basic Law V will settle the relevant identity in question. The neo-logicist,969

on the other hand, in rejecting the recipe—and a single domain of primitive970

objects generally—in favor of a multitude of distinct abstraction principles de-971

scribing distinct (yet possibly overlapping) domains of mathematical objects,972

suffers from no such arbitrariness. But the cost of avoiding the arbitrariness973

found in Frege’s project is the C−R problem.974

975

roy t. cook976

University of Minnesota977

philip a. ebert978

University of Stirling979

980

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